Alexandrite is the color-change variety of chrysoberyl, beryllium aluminum oxide, formula BeAl2O4. It is orthorhombic, hardness 8.5, and usually forms tabular or cyclic twinned crystals. The color change is real: chromium-bearing chrysoberyl can look green in daylight and red to purplish in incandescent light because those light sources emphasize different parts of its spectrum.
Ordinary yellow or green chrysoberyl without a clear color change is the same species, but it is not alexandrite. This page is only for the color-change variety.